The modem/wireless router signal is audible in most all the tube amps I have on the bench when turned up a lot. I know for sure this is true because it stops when I disconnect the power to the modem. I ASSUME the frequency from the modem/wireless router is somewhere 2 to 5GHz. How that produces an audible signal in the amp I am not sure but it does.

I assume brass or copper screen can be had from somewhere if not locally. If I can find some I guess I just build a box out of the wire to contain the chassis and ground it to the chassis??

Can I build some sort of metal/wire mesh container to place over an amp?

Yes you can, but it is probably not worth the effort. We have all had those thoughts, some of us have even built Faraday screens and then found it impossible to access the amplifier with external cables, without letting that demonic RFI back in.
Where an RFI free space is really needed you will have to build a screened room. That requires two separate conductive walls, one inside the other. It also requires screened access, ventilation and power supplies.

Blaming some unexpected amplifier behaviour on invisible RFI or “spikes on the mains” is just too easy. Avoid going on a “witch hunt” for a hidden source of RFI that is just beyond the range of the available instrumentation or outside your technical understanding. If moving your body near the amp influences the “symptom” then it may be caused by RFI. If it is not RFI then it is probably poor amplifier design. An amplifier should be designed to keep it's internal signals from feeding back on each other. That internal “self immunity” should also screen it from stray external influences such as RFI.

The human species would not have survived to design amplifiers today if it did not have just enough paranoia to avoid threats and so stay alive. We are all a little bit paranoid which keeps us alive and still able to find explanations by the application of scientific methods. A good engineer is cautious, with just enough paranoia to avoid accidents in design. Without some paranoid caution an engineer would be dangerous.
You cannot be sure you are a truly good person unless you have bad thoughts and resist them. Likewise, we cannot be good engineers unless we question all possible influences on our designs, then rationally allow for every one of them. Maybe that is why the portraits of great engineers always seem to show them looking a little bit worried.

Remember that the speaker leads also acts as an antenna, injecting signals in the amplifier through the feedback. The tube getting the feedback cannot follow the high frequency, so the grid - cathode works as a diode. Thus you have an AM detector in your system, and what you hear is the demodulated signal.

The usual fix is to introduce an inductor in the "live" wire to the speakers. Get hold of some thick, lacquered copper wire and wind it on a pencil for about one inch in length. Use this coil in parallel with a 22Ω resistor close to the output transformer, but on the "outside" of the feedback connection.

While I was thinking about all this "cage building" I was wondering how I could deal with all the "antennas" hooked up to the amp. The guitar, the cable, the speaker and wire, and the power cord. It did not look like it would be easy to deal with if it could be done at all.

As far as I can tell, most if not all the issues I am experiencing started when AT&T installed new cable service. Without question the AT&T modem/router is producing audible sounds in most all amps on the bench. Of course I can and do turn off the device....a bit of a pain, because that turns off the internet, the TV, the wifi connection to my cell phone and the VOIP office phone.

There is another issue I am experiencing. Some amps plugged into the power on my bench exhibits more "hum" than when I take the amp to another building in another part of the city.

Better amp design is a key methodology to dealing with some of these issues. As I am frequently repairing vintage amps from the sixties and building reproductions of those amps I don't have much control over those designs.

I think the next step is to address the mains power and perhaps buy some sort of power conditioning device. I don't see a ground rod of the type I see in other parts of the country (3/4 inch copper rod 8 foot long) connected at the main panel here. Because there is a lot of very hard rock here, it may not be very easy to install such a grounding rod.

There is another issue I am experiencing. Some amps plugged into the power on my bench exhibits more "hum" than when I take the amp to another building in another part of the city.

That's interesting.
Notice they always place output transformers away from power transformer so as to minimize magnetic coupling between them ?

Have you ever looked at leakage flux from a transformer ?
A small coil hooked to your oscilloscope is a flux detector
because e = -nturns X Δflux / Δtime

Placing such a coil adjacent a transformer core gives you a real peaky wave because more flux "leaks" out of the core wen it's well up the saturation curve, at peak of magnetizing current's rough sine-like wave.
I used such a flux detector in the power plant. SInce you're experimenting, try it. You can wind a coil or steal one from a small relay or solenoid. more area makes it more sensitive but more unwieldy too. Try several and see what works.

Leakage flux from a power transformer will be a strong function of line voltage. Would be interesting if you found hum related to voltage variations around town. If not, well, you know one more thing it isn't.

One thing that WILL vary around town is RF interference from 60 Hertz arcing on dirty insulators and things of this nature. This may show up in an amplifier as hum if the arcing is severe enough. Some amateur ops that I used to know always liked a good hard rain. It would wash the dirty insulators off and relieve them of what was referred to as 'line noise'.

Have you ever looked at leakage flux from a transformer ?
A small coil hooked to your oscilloscope is a flux detector
because e = -nturns X Δflux / Δtime

Reminds me: A client had once had problems with electronic noise interfering with their circuit. When they disconnected "earth" (well, the thing was supposed to go underwater in a stainless steel encasing...) it sort of solved the issue, but their customer flatly denied to accept the product with that "fix". I was called in to look at the electronics. In order to determine the source of the noise, I took a wire about 30 cm long, made a loop and connected it between "common" and "live" on an oscilloscope probe. Bingo! Large spikes! Moving the loop around, the problem was located at the switching supply. The construction was OK, but the layout was horrible...

I took a wire about 30 cm long, made a loop and connected it between "common" and "live" on an oscilloscope probe.

Great technique.
I used a ten turn coil of area 1/10 square meter to estimate 60 hz sinewave fields around the power plant. That simplifies conversion from volts to milliTesla . I shoulda incorporated pi in the area because ω = 2πf.

I reinstalled all the wiring to my bench. That did reduce the hum in a amp I am working on. I also moved all test equipment to one circuit with nothing else on that circuit except the amp being tested. I did some testing with the computer system here on the bench and it does not seem to be implicated in any issues.

I am going to give Jim's device a try.

Some of this "noise" issue in amps is a bit hard to understand. I have had two identical 1965 Fender Deluxe Reverb amps side by side with one being dead quiet and the other producing a bit of hum. Both played correctly and nothing seem to change anything much, like changing tubes, resoldering, changing out suspect components.

Part of the issue for me is that I do not have a regulated bench supply to do testing with. In years past high voltage regulated supplies were common. Not today. I am looking at building one but I am not sure I have the skill to do it correctly. Screwing up something putting out 700 plus volts could cause some exciting conditions...lol

The most perplexing issue with all this relates to one particular amp I built. In the last few months I have built ten or eleven new amps which worked as I expected without any issues. The one in question has given me nothing but problems. I guess at this point I am going to take it completely apart and start over.

The one in question has given me nothing but problems. I guess at this point I am going to take it completely apart and start over.

Divide and conquer . Try powering it from a different transformer ? Can you pull the 5U4 (or whatever) and patch B+ from another amp ?

Computer guys use a method called "Binary Search" , where you cut your search area in half . It's said to be fastest way. But in hardware there's seldom a convenient exact 50% point where you can split it. So i'd ask "Transformer or amp? " A split at 1/e , 1/2.718 . 63/37 % probably works as well as 50-50.